A Quantized-Diffusion Model for Rendering Translucent Materials

Abstract

We present a new BSSRDF for rendering images of translucent materials. Previous diffusion BSSRDFs are limited by the accuracy of classical diffusion theory. We introduce a modified diffusion theory that is more accurate for highly absorbing materials and near the point of illumination. The new diffusion solution accurately decouples single and multiple scattering. We then derive a novel, analytic, extended-source solution to the multilayer searchlight problem by quantizing the diffusion Green’s function. This allows the application of the diffusion multipole model to material layers several orders of magnitude thinner than previously possible and creates accurate results under high-frequency illumination. Quantized diffusion provides both a new physical foundation and a variable-accuracy construction method for sum-of-Gaussians BSSRDFs, which have many useful properties for efficient rendering and appearance capture. Our BSSRDF maps directly to previous real-time rendering algorithms. For film production rendering, we propose several improvements to previous hierarchical point cloud algorithms by introducing a new radial-binning data structure and a doubly-adaptive traversal strategy.

Octoguy model and textures courtesy of Bradford deCaussin.

Contributions

The first recognition in graphics of the relationship between BSSRDFs and the searchlight problem in transport theory (and related literature).

The proposal (following optics) of a notation for volume properties that clearly distinguishes between material cross-sections and material coefficients (since both appear in parametric tissue models, for example).

The first formal study in graphics of the limits of diffusion theory for highly absorbing materials and near sources and boundaries.

The introduction to graphics of the idea of modified diffusion theory, including the evaluation of many different alternative forms and selection of one (Grosjean’s) that seemed most appropriate for use in computer graphics.

The first application of Grosjean’s modified diffusion theory to method-of-images BSSRDFs producing an accurate decoupling of single and multiple-scattered light.

The first recognition in graphics that the diffusion boundary conditions being used in diffusion BSSRDF models are not actually satisfied by mirroring negative sources about the extrapolation plane outside the medium (an error inherited from the medical physics literature).

The first application in graphics of asymptotically consistent diffusion boundary conditions for reflecting boundaries.

The first application in graphics of an exitant-flux calculation consistent with the full form of the diffusion solution at the bounary (Kienle and Patterson’s).

The first application in any field of an extended diffusion source model together with Kienle and Patterson’s exitance formulation, which is essential in order for the extended source model to outperform the dipole/multipole models.

Introduction of a new analytic BSSRDF model for slabs and multilayer materials using an ‘extended multipole’: a generalization of the multipole model for slabs by mirroring continuous exponential (beam) source distributions about extrapolated boundaries. Additionally, we evaluated the accuracy of this model for quite thin material layers and included a new reduced-intensity term to the adding-equations method for combining layers.

Introduction of a closed-form variable-accuracy quadrature for the extended (beam) source multiple-scattering component of the BSSRDF via temporal quantization of point source diffusion Green’s function. This produces radial profiles that are sums of Gaussians, providing a new physical foundation to a previous popular expansion of BSSRDFs for real-time rendering (and thereby avoiding the necessity for any non-linear fitting procedures to find such Gaussian expansions).

A new efficient, stable and energy-conserving method for convolving two layer profiles when the inputs are expressed as sums of Gaussians.

Introduction of a missing normalization term in the angular/spatial-factored expression of diffusion BSSRDFs.

A new ‘radial binning’ acceleration method for exploiting the radial symmetry for the [Jensen Buhler 2002] algorithm.

A new doubly-adaptive acceleration method for reusing similar far contributions for the [Jensen Buhler 2002] algorithm.

PDF (free download from ACM)

Back Story

I sometimes get asked how this research came about and how I came to discover the related work. Here’s what I can remember:
[Donner and Jensen 2005] cited Tuchin’s book on Tissue Optics, which I decided to buy a copy of May 1, 2007 (a date that amazon.com still recalls).
There is a single sentence on p.15 of Tuchin, which I don’t recall noticing until early 2010 in Wellington: “Some limitations of the diffusion theory, in particular connected with bad description of the fluence rate if one gets to the source, can be gotten over when it is modified on the basis of accurate but simple Grosjean’s equation, which describes the light distribution in infinite isotropically scattering turbid media [Graaff and Rinzema 2001]”. It was a surprisingly important statement with no followthrough in the book, so I quickly starting chasing the trail.

I remember reading [Graaff and Rinzema 2001] on a flight to Queenstown in Feb 2010. It was quite a helpful paper to understand Grosjean’s work at first, but in the end I was short on space and didn’t cite it, as the paper itself didn’t have anything new that I required.

At this time I had MCML making many profiles and the goal was simply to find better equations to fit the MC output. A few papers down in Tuchin’s bib was [Kienle and Patterson 1997] which I have a written checkmark next to, so I must have been downloading all the related papers there.

I wrote a modular BSSRDF toolkit in Mathematica that let me plug in different point source green’s functions and use either discrete dipole or extended-integral source functions and general exitance calculation methods. I vaguely recall batching a bunch of different combinations and noticed that Grosjean + extended-source + KienlePatterson was a unique winner. And I couldn’t find anything citing KP97 that had tried it.

But now I was stuck with the annoyance of the integral. And I knew I wanted Gaussian sums from my previous work and the idea I was still clinging to that dated back to 2007, which was a layer convolution in the space of Gaussian sums as polynomial multiplication that avoided any ringing issues with the Hankel transforms (but the fast version of the convolution was all Geoffrey – took him less than an hour to find the best algorithm).

I tried to express the diffusion green’s function as an integral of Gaussians at some point and failed. It was only later, reading more papers on modified diffusion to see if I could do even better than Grosjean, that a paper by Razi Naqvi “On the diffusion coefficient of a photon migrating through a turbid medium: a fresh look from a broader perspective” made it very clear: the steady state green’s function had a temporal decomposition as an integral of Gaussians. It was fairly easy from there to discretize the integral into a quadrature of sorts to get Gaussian sums back on the surface when using the extended source. Also didn’t cite Naqvi in the end, the one equation I needed dated back to Patterson in 89.

Of course, the work stands heavily on that of Henrik and others for making clear the visual difference in a translucent render and connecting and extending diffusion methods from other fields. Henrik attributes the discovery of the related work on diffusion dipoles to Bohren’s book ‘Clouds in a Glass of Beer’, which has a section on Dipoles, but doesn’t cite or specifically talk about the diffusion dipoles.

Errata, Discussion and Reviews

We include additional information about stable numerical evaluation of the model here:Publon 2802

Original Score

Music composed for the Fast Forward program at SIGGRAPH 2011, Vancouver: